In this section, we observe that non-linear predictors can surpass the lower bound (REF ) that holds for linear predictors in \(\mathcal {W}_{b}\) . In particular, via known results in the literature, we first demonstrate that the Vovk-Azoury-Warmuth (VAW) forecaster yields an exponential improvement on the boundedness constants \(b\)  and \(r\)  compared to the lower bound (REF ). We remark that in our setup, the VAW forecaster was previously used to match the lower bound (REF ) in [1]}; the difference in the result below is that we tune the regularization parameter and use the resulting upper bound to demonstrate a statistical separation between proper and improper algorithms, rather than matching a lower bound that holds for proper algorithms. Finally, we discuss a less known modification of the VAW forecaster due to Forster and Warmuth [2]} that can completely remove the dependence on the boundedness constants \(b\)  and \(r\)  and removes any assumptions on the distribution of the covariates.
